Triangle similarity theorems

Before trying to understand similarity of triangles it is very important to understand the concept of proportions and ratios, because similarity is based entirely on these principles.Triangle similarity is another relation two triangles may have. We already learned about congruence, where all sides must be of equal length. In similarity, angles must be of equal measure with all sides proportional. Similarity is the relation of equivalence.
Two triangles and are similar, thus we write: .

There are four theorems that we can use to determine if two triangles are similar.

AA theorem

Two triangles are similar if their two corresponding angles are congruent.

Let be the given triangle. So how can we construct a similar triangle?

We will expand segment lines and over the vertices and , respectively. On the line we choose the point and construct a line that is parallel to line and that passes trough a point . The intersection of the previously constructed line and line is point .

The resulting triangle is similar to the given triangle , as shown below.

(if two corresponding angles are of equal measure, then the third is also equal and corresponding).

The following proportions are also true:

.

SSS theorem

Two triangles are similar if the lengths of all corresponding sides are proportional.

Triangles and are the similar triangles if:

Example 1.

Are the following triangles

similar?

Solution:

We have:

It follows that triangles and are thus similar by the SSS theorem.

If we found a different value in any part, the triangles are not similar.

SAS theorem

Two triangles are similar if the corresponding lengths of two sides are proportional and the included angles are congruent.

Triangles and are similar if and . It follows that all corresponding angles are congruent and the lengths of all sides are proportional.

Example 2.

Are the following triangles

similar?

Solution:

It must be: .

We have:

These two corresponding sides are proportional and the included angles are of equal measure. It follows that the triangles and are thus similar triangles according to the SAS theorem.

SSA theorem

Two triangles are similar if the lengths of two corresponding sides are proportional and their corresponding angles across the larger of these two are congruent.

Consider triangles and . If we know that and if the angles across the larger ones are congruent, then triangles and are similar.

Example 3.

Are the following triangles

similar?

Solution:

We must compare all sides by their lengths.

In this case we have

which is true. The opposite angle to the side of the longest length in triangle is and opposite angle to the longest side in triangle is .

It follows that , which means that triangles and are thus similar by the SSA theorem.